Try restaurants that get produce in large flat plastic crates/boxes.
I have one I got (part time job with Chic-Fil-A gives me access) at work that is
about 6-8 inches deep and maybe 30 inches long....I have used it so many times
this season to cart plants around in and it is washable and open for air.
Love it. It would work for your needs and they are very sturdy. I
also have a cardboard one that I was going to dispose of but like you mentioned,
it is harder to find these kind of things, so I may hang on to it as well.

We had over 250 varieties for the sale this year. The club gets
together to clean all the irises and mark the leaves with the price and
names. It takes about 5 hours each day. One of the local nurseries
went out of business and gave us her plastic plant trays. These are used
both to dry the irises and display them at the sale. It saves the
step of transferring irises from the window screens to cardboard flat
boxes. The bars are not using boxes anymore for beer. They are
shrink-wrapping beer. Last year we had to scrounge to find enough
cardboard flat boxes. The plastic trays are much nicer.

I usually put a plastic chair in the shade
surrounded by three plastic garbage cans of water plus a 5 gallon bucket
with a cup of Clorox. Use one garbage can for the first rinse, next one
for second rinse, then a brief Clorox dip, then a final rinse in the 3rd
garbage can, and spread them out on old window screens in the shade to
dry.

I haven't had a lot of donations to give to our club sales
recently - mostly just growing seedlings - but back in the day, that was
usually enough washing to get a few hundred rhizomes quite
clean.

Fortunately, my soil is organic gravelly loam, so other than
bits of sand or small rocks and a fine layer of soil, most soil just
shakes off and a quick swish thru a bucket of water gets them pretty
clean. Water that sloshes on the ground disappears fast on my gravelly
creek soil. & it's usually hot enough at digging time that the
spring water feels good to play in.